Question

What is the correct form of the partial fraction decomposition for the expression (7x+18) / (x^2+9)?

a. A/x^2 + B/9x
b. A/x + B/x+9
c. (Ax + B)/x^2 + C/9x
d. (Ax + B)/x + C/(x+9)

Answer 1

The correct form of partial fraction decomposition for the expression (7x+18)/(x^2+9) is (Ax + B)/(x^2) + C/(9x).

Step-by-step explanation:

The correct form of partial fraction decomposition for the expression (7x+18)/(x^2+9) is option c. (Ax + B)/(x^2) + C/(9x). To decompose the rational expression, we need to find the values of A, B, and C. We can do this by equating the numerator of the original expression to the numerator of the decomposed expression. We get the equations 7x + 18 = (Ax + B) + C(x^2).

By comparing coefficients of similar powers of x, we can solve for A, B, and C. In this case, A = 0, B = 7, and C = 2.

Answer 2

The correct form of the partial fraction decomposition for the expression (7x+18) / (x^2+9) is (Ax + B)/(x^2+9), since the denominator is a quadratic with no real roots and can't be split into real linear factors.

Step-by-step explanation:

The correct form of the partial fraction decomposition for the expression (7x+18) / (x^2+9) is not directly given in options a-d, but we can determine it based on the denominator. Since the denominator x^2+9 is a quadratic with no real roots, the partial fraction decomposition must take the form of (Ax+B)/(x^2+9). This is because partial fraction decomposition splits the fraction into simpler parts and since x^2+9 cannot be factored into real linear factors, we leave it as is, only factoring out an x term if it were present in the numerator which is not the case.

Options a and b are incorrect because they Suggest decomposing into linear or repeated linear factors which don't match the original denominator. Option c mistakenly includes C/9x indicating a potential repeated linear factor of x which is not present in x^2+9. Therefore, by process of elimination and understanding of partial fractions, we must adjust option d to (Ax + B)/x^2 + C/(x^2+9), but since C/(x^2+9) would be redundant, we simply use (Ax + B)/(x^2+9).